Homework 05 : The Morse Potential

The Morse potential for λ = 1.

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-1- The Morse Potential

To model the interaction of two atoms in a diatomic molecule, Philip M. Morse proposed the following potential:

$V(x) = \frac 12 \lambda^2 \Big(1-e^{-x}\Big)^2$

It is one of the few analytically solvable models of quantum mechanics: its eigenvalues are given by

$E_n = -\frac{1}{8}(2n+1)(2n+1-4\lambda) \quad n=0,1,\dots$

and the eigenfunctions by

$\psi_n(x) \propto z^{\lambda - n - \frac 12} e^{-\frac 12 z}L_n(z;2\lambda-2n-1) \quad\text{ where }\quad z=2 \lambda e^{-x}$

where Ln(z;α) is a Laguerre polynomial which expresses as:

$L_n(z;\alpha)=\frac{z^{-\alpha}}{n!}e^{z}\frac{d^n}{dz^n}(e^{-z} z^{n+\alpha})$

In this exercise, we recover some of these non-trivial results with the density matrices and path integrals.

1. Determine analytically the ground state ψ0(x) and the first excited state ψ1(x) up to a constant factor.
2. Determine explicitly the normalisation of these wavefunctions. What happens if λ is too small? Comment. In particular: discuss in details the number of bound states , especially in relation (i) to the expression of the eigenvalues given above and (ii) to the shape of the Morse potential.

-2- Density matrix approach to the Morse potential

1. Produce a few pictures of the Morse potential, for different parameters λ > 3/2.
2. Justify that one can restrict the space to some finite interval.
3. At high temperature, set up the density matrix as a numpy array, on a grid of points chosen with numpy linspace.
4. Perform the matrix-squaring procedure using the numpy.dot product. Plot the entire density matrix using matshow (part of pyplot in matplotlib) , and explain the two-dimensional figures for different temperatures, especially the "width" of the density matrix.
5. At low temperature, and for several values of λ, compare:
   • (i) the diagonal density matrix, determined numerically from the matrix squaring algorithm, and
   • (ii) the known expression of those diagonal elements in terms of the groundstate wavefunction (see equation (3.5) in SMAC keeping only the ground state n=0 in the sum).
6. Explain why we can say that «the temperature is low» when those two quantities are equal.
7. A good choice of values for parameters is N=100 slices in space, λ = 2, βinitial = 2−6, 9 iterations of the matrix squaring, with restricting r to the interval [-2,10]. Find other values of the parameters yielding consistent results.

-3- Path-integral simulation for the Morse potential

Set up a naive path-integral simulation (as in SMAC algorithm 3.4) for a single particle in the Morse potential. Take a moderate number of time slices, and produce a histogram of the particle positions. Explain the exact relationship of this histogram with the diagonal density matrix. Again compare with the analytic solution at low temperature.

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